A controllable system:

An uncontrollable system, since there is no way to affect the second state:

Test the controllability of a system with approximate coefficients:

Exact coefficients:

Symbolic coefficients:

Multiple-input system:

Discrete-time system:

A descriptor system:

Controllability is equivalent to controllability of both slow and fast modes (C-controllability):

Test controllability of individual eigenmodes:

The system is uncontrollable because of eigenmode :

This can be seen in the Jordan form, where there is no way to affect the third state:

Test controllability of an AffineStateSpaceModel:

If an operating point is given, controllability from is tested:

The system is controllable from a generic point:

The position and velocities of all three masses can be controlled by the force on :

An electric circuit with the capacitor voltage and inductor current as states:

In general, the system is controllable:

However if , it is not controllable:

A unicycle with drive and steering as input:

The system is controllable, but the linearized system is not:

The parking problem:

The system is controllable, but the linearized system is not:

A rotating rigid satellite modeled using Euler's equations of motion:

Its controllability for various values of the principal moments and actuator combinations:

A grid to show the results:

If , the system is controllable with any two actuators:

If , the system is not controllable with actuators 1 and 2 or any one actuator:

If , the system is controllable only with all 3 actuators:

A diagonal system is controllable, assuming and :

With , there is no way to control the first state:

With , the first state cannot be controlled directly, but indirectly from the second state:

With , the second state cannot be controlled directly or indirectly from the first state:

Use JordanModelDecomposition to compute the canonical state-space representation above:

Compute the controllability of each mode using the "PBH" test:

For descriptor systems, KroneckerModelDecomposition is the generalization of the diagonal form:

Determine the controllability of the slow subsystem from its structure:

Compute it using the original system:

Determine the controllability of the fast subsystem:

Compute it using the original system:

If the descriptor matrix of a StateSpaceModel has full rank, there is no fast subsystem:

Hence the complete controllability of the system can be evaluated from the slow subsystem:

Controllability is invariant under a nonsingular StateSpaceTransform:

Controllability is invariant under state feedback:

Compute the state feedback using StateFeedbackGains:

The closed-loop system is also controllable:

Controllability does not mean it is output controllable (OutputControllableModelQ):

Output controllability does not imply controllability:

The Gramian method is not reliable for systems that are not asymptotically stable:

The eigenvalue in the right half of the complex plane leads to instability for continuous-time systems:

For affine systems with nonzero drift, the distribution tests only for accessibility:

A random initial condition and random input signal generator:

Simulate the system 10 times with random inputs:

The system cannot be moved to the left of the initial point: